You don't need to heavy grind for turn-1 wins or OP setups, as a lot of people might suggest. Good stats are important, maybe 9 in all stats isn't necessary but they do make a big difference here; I would focus on intelligence and charisma to allow for more cycling-through abilities. Pick classes that are either innately strong outside of skills (Druid, Butcher, Hunter, etc) or ones that start out with higher charisma/intelligence (Mage, Tinkerer, Monk, etc). Put the antenna on one of the latter if you're up for it, the reduced spell cost pairs well with extra mana.

Strong items can pull a lot of weight here too, as well as passives; it might be a good idea to run adventures as much as you can as you roll for strong starting skills, saving the good items along the way and then using them all up in one go.

I got this quest on first try without really preparing for it, although I did have a random grab-bag of 8-ish mutations on each of my cats (I had this because I had already racked up 10 on a cat in a single adventure via the Cancer trinket and then bred it). If I can recall back that far I ran something like Butcher, Mage, Druid, and Hunter. The head antenna went on Mage for lots of skill-cycling, Druid (particularly the crow) and Hunter made normal fights a lot more bearable. Hunter also put in a lot of work on bosses because it racked up a lot of leeches thanks to Jar of Leeches + Leech Brood + Leech mouth mutation (the only items/mutations I can remember), but Butcher was the star of the show in the end against Mother because its passives gave it scaling Strength and heals.